Optical excitation of phase modes in strongly disordered ...

Optical excitation of phase modes in strongly disordered superconductors T. Cea,1 D. Bucheli,2 G. Seibold,3 L. Benfatto,1 J. Lorenzana,1 and C. Castellani1 1

ISC-CNR and Dep. of Physics, “Sapienza” University of Rome, P.le A. Moro 5, 00185, Rome, Italy ISC-CNR and Dep. of Physics, “Sapienza” University of Rome, P.le A. Moro 5, 00185 Rome, Italy 3 Institut F¨ ur Physik, BTU Cottbus, PBox 101344, 03013 Cottbus-Senftenberg, Germany (Dated: January 16, 2014)

arXiv:1401.3637v1 [cond-mat.supr-con] 15 Jan 2014

2

According to the Goldstone theorem the breaking of a continuous U(1) symmetry comes along with the existence of low-energy collective modes. In the context of superconductivity these excitations are related to the phase of the superconducting (SC) order parameter and for clean systems are optically inactive. Here we show that for strongly disordered superconductors phase modes acquire a dipole moment and appear as a subgap spectral feature in the optical conductivity. This finding is obtained with both a gauge-invariant random-phase approximation scheme based on a fermionic Bogoliubov-de Gennes state as well as with a prototypical bosonic model for disordered superconductors. In the strongly disordered regime, where the system displays an effective granularity of the SC properties, the optically active dipoles are linked to the isolated SC islands, offering a new perspective for realizing microwave optical devices. PACS numbers: 74.20.-z,74.25.Gz, 74.62.En

I.

INTRODUCTION

In the last decades the failure of the BCS paradigm of superconductivity in several materials led to a profound modification of the description of the superconducting (SC) phenomenon itself. A case in point is the occurrence of Cooper pairing and phase coherence at distinct temperatures, associated respectively with the appearance of a single-particle gap ∆ and a non-zero superfluid stiffness Ds . This behavior is observed, e.g., in high-temperature cuprate superconductors1,2 , strongly-disordered films of conventional superconductors3–8 and recently also in SC heterostructures9 . In all these materials the BCS prediction that Ds is of order of the Fermi energy, much larger than ∆ ∼ Tc , is violated due to the strong suppression of Ds . The resulting scenario, supported by systematic tunneling measurements, suggests that pairing survives above Tc , leading to a pseudogap state dominated by phase fluctuations enhanced by the low Ds value.10 In all this, optics represents a preferential playground to address the peculiar role of disorder. Indeed, as we show in this Communication, disorder renders collective modes - optically inactive in a clean superconductor visible. By analyzing a prototype fermionic model, the attractive Hubbard model with on-site disorder11–16 , we reveal that thanks to the breaking of translational invariance the collective modes couple to light via an intermediate particle-hole excitation process. Most remarkably, this coupling leads to the emergence of additional optical absorption, mainly due to phase modes, below the BCSlike threshold for a photon to break apart a Cooper pair, in agreement with recent experimental observations17,18 . Deeper insight into the nature of this disorder-induced optical response is then gained through a comparison with the XY model in transverse random field. Within this effective bosonic description of disordered superconductors19,20 we show explicitly how the local inhomogeneity of the superfluid stiffness leads to a finite

electric dipole for the phase modes. At strong disorder, where the system segregates into SC islands of tens of nanometers,5,8,15,16 the SC dc current flows along preferential percolative paths through the good SC regions16 . As a consequence the finite-frequency optical absorption occurs in the remaining isolated SC regions, thanks to the presence of a finite phase difference between the opposite sides of the islands, which then act as nano-antennas. This nano-scale selective optical effect, that we propose to test via microscopic imaging21 , can be used to tune the resonant frequency and the quality factor of superconducting microresonators.22 II.

FERMIONIC MODEL

The model Hamiltonian we consider to investigate a disordered superconductor is the attractive Hubbard model (U < 0) with local disorder Vi ∈ [−V, V ] and hopping t restricted to nearest-neighbors, X † X X H = −t (ciσ cjσ +h.c.)+U ni↑ ni↓ + Vi niσ (1) hijiσ

i

iσ

which we solve in mean-field on a N ≡ Nx × Ny lattice (up to N = 20 × 20 with periodic boundary conditions) by using the BdG approach23,24 . The total current in direction α is defined as usual as: Jα (q, ω) = −e2 Kαβ (q, ω)Aβ (q, ω), (2) Kαβ (q, ω) = Dδαβ − χαβ (q, ω) . (3) D E
P Here D = −t c†n,σ cn+ασ is the diamagˆ + h.c. n,σ N netic term, where h. . . i denotes the thermal and disorder average, which restores the translational invariance for model (1), allowing one to define the Fourier transform χαβ (q, ω) of the correlationh function for the paramagi P † netic current jnα = −it σ c†nσ cn+α,σ ˆ − cn+α,σ ˆ cnσ . In

2

2Δ!

(b) !"%$

0.8

0.8

0.6

0.6 mS/mN

mS/mN

!"#$

(a)

0.4 0.2

(c)

0

0.4 0.2

0 1 2 3 4 5 6 7 8 t/6

0 0 1 2 3 4 5 6 7 8 t/6

FIG. 1: Schematic of the optical absorption σS /σN (S=SC, N=normal state) in a disordered superconductor. In the BCS approach (left) only the single-particle excitations across the SC gap 2∆ (a) are included, corresponding to the bare-bubble approximation (b) for the current-current response function. The resulting optical conductivity (c) consists of a delta peak at ω = 0 of weight DsBCS (arrow) plus a regular part (solid line) starting at ω = 2∆. When vertex corrections are included (right) an excited quasiparticle can be converted in a collective mode (a), described in the diagrammatic approach (b) by the RPA resummations of the corresponding amplitude, phase or density fluctuations. An additional absorption appears at energies ω < 2∆ (c), corresponding to a superfluid peak at ω = 0 with strength Ds < DsBCS .

a superconductor the superfluid stiffness is defined by the transverse q → 0 limit of Eq. (2). For example, for a field along the x direction one has Jx = −e2 Ds Ax where Ds = D − Re χxx (qx = 0, qy → 0, ω = 0).

(4)

The optical conductivity is obtained from Eq. (2) by assuming a homogeneous vector potential, so that Ax (ω) = Ex (ω)/iω and the real part of the optical conductivity is xx (q=0,ω) σ(ω) = −e2 Re Ki(ω+i0 + ) , leading to σ(ω) = e2 πδ(ω) [D − Re χxx (0, ω)] + e2 ≡ e2 πDs δ(ω) + σreg (ω)

Im χxx (0, ω) ω (5)

where we separeted explicitly the superfluid response at ω = 0 from the regular part σreg occurring at finite frequency. By using the Kramers-Kronig relations for the χxx one then finds the well-know optical sum rule Z ∞ Z ∞ πe2 πe2 dωσ(ω) = Ds + σreg (ω) = D. (6) 2 2 0 0+ The above Eq. (6) shows that any paramagnetic process described by σreg leads to a suppression of the superfluid stiffness with respect to the diamagnetic term D, which at small density and weak interactions reduces to the usual form D ' n/m. In the BCS theory χ ≡ χ0

is computed in the so-called bare-bubble approximation (see Fig. 1b, left)23 , in which one includes only particlehole excitations on top of the BCS ground state. At T = 0 these excitations are exponentially suppressed by the opening of the gap, so that the optical absorption is possible only above the threshold to break a Cooper pair, i.e. at ω > ωpair = 2∆ (see Fig. 1c, left). Provided that ωpair is smaller than the inverse lifetime of quasiparticles BCS the resulting σreg (ω) is given by the well-known MattisBardeen formula25 , and the superfluid stiffness DsBCS is smaller than D already at T = 0. In the following we will show that also collective modes, neglected in the BCS approach, give rise to a finite contribution to σreg (ω) at strong disorder, that is located mainly below ωpair (see Fig. 1c, right). This additional optical absorption is accompanied by a further reduction of Ds with respect to DsBCS 16 , that has been experimentally reported4,26,30 . The full optical response beyond BCS level can be computed by including vertex corrections23 , that also guarantee full gauge invariance of the theory23,27 . The currentcurrent correlation function χ can then be expressed in a compact form as (see App. A): ˆ T V [1 − Π ˆ 0 V ]−1 Λ ˆ χ = χ0 + Λ

(7)

ˆ is the vector containing the correlation funcwhere Λ tions that couple the current jnα to collective modes, i.e. particle-particle (amplitude and phase) and density fluctuations, described by the RPA resummation of the bare ˆ 0 , see Fig. 1b right. The Vˆ and Π ˆ 0 are susceptibility Π matrices both in real space and in the phase space of collective modes, and translational invariance for χ is recovered after average over disorder configurations. In the clean case collective modes contribute only to the longitudinal response at finite q23,27 . In contrast, disorder ˆ susceptibilities finite even for a q = 0 exterrenders the Λ nal perturbation, so that the collective modes contribute to the optical response. Notice that this optical mechanism is similar to the one discussed recently for few-layer graphene28 to explain the huge infrared-phonons peaks29 . In that case doping activates the intermediate particlehole process, analogously to what disorder does in our problem. The results for the optical conductivity at finite frequency for two representative values of coupling U and disorder are shown in Fig. 2, along with their BCS counterparts. As one can see, the major differences between the two appear below the scale ωpair = 2∆, marked with a dashed line. Notice that in the model (1) the spectral gap ∆ in the single-particle excitations remains finite (and relatively large) at strong disorder, as it has been discussed previously11,12,15 . As a consequence, the BCS calculation always shows a finite threshold at ωpair , with a profile that coincides at low disorder with the Mattis-Bardeen prediction25 . In contrast, the full response extends also below ωpair , with a shape and intensity that depend both on the SC coupling U and disorder. This result can explain the residual optical absorption in the microwave regime17,18 and deviations from BCS

3

0.2

U/t=2 0.1 V/t=0.5

σ(ω)/σ0

0.4

0

0

0.15 0.1

0.5

1

4

U/t=5 V/t=0.5

0.05 6

0

0

U/t=2 V/t=3

0

ω/t

0.2 0.4

10 0.04

0.02

5

0

5

0.1

0.04

0.05 0

0

2

0

0

0.05

0.1

0.2

0

σ(ω)/σ0

0.1

0.15

0.05

0.02

0.05 0

1

2

3

10

0

U/t=5 V/t=3 0

0

5

10

ω/t

0

1

2

15

FIG. 2: σ(ω) in units of σ0 ≡ e2 /~ for the fermionic model (1). Here N = and we averaged over ... disorder configurations. The main panels report the curves without (thin, red) and with (thick, black) vertex corrections, while the dashed vertical lines mark ωpair = 2∆. Insets: zoom of the lowenergy part along with the results of the bosonic model (8) (dashed lines), with W, J parameters assigned as described in the text.

FIG. 3: Comparison between σ(ω) computed using the full gauge-invariant response (solid line, black) and the contribution of phase fluctuations only (dashed line, red) (see Appendix A). The results of Fig. 2 for the bosonic model (8) (dotted line, blue) are reported as well.

bosonic model for the disordered superconductor, i.e. the XY spin 1/2 model in a transverse random field19 :

HP S ≡ −2

X

ξi Siz − 2J

i

theory4,26,30 observed recently at strong disorder. In particular, the smearing of the ωpair treshold due the presence of a dissipative channel associated to phase modes can lead to an apparent optical gap smaller than the one measured by STM, explaining the puzzling results of Ref. [18]. At the same time this effect can influence the performance of superconducting microwave devices, a field that has grown dramatically over the past decade22 . Finally, two remarks are in order with respect to the results of Fig. 2. First of all, we checked that even though all the collective modes enter in the full response, the main contribution to σreg (ω) at low energy stems from phase fluctuations. This is shown explicitly in Fig. 3, where σreg (ω) has been computed by including only the phasecurrent vertex in Eq. 7 (see Appendix A). Second, one could wonder what happens when the Coulomb interaction, neglected in the present calculations, is taken into account. Indeed, one usually expects that the presence of long-range interactions the sound-like dispersion of phase modes is converted into a plasmonic one.27 However, we do not expect that this result will spoil our conclusions: indeed, while plasmonic modes appear in the longitudinal response, the transverse optical response, which is the one discussed here, should be anyway screened, as shown explicitly in the weakly-disordered case in Ref. [31].

BOSONIC MODEL

To systematically address the structure of the phase excitations responsible for the sub-gap absorption we compute the optical conductivity also within an effective


Si+ Sj− + h.c. .

(8)

hi,ji

In the pseudo-spin language S z = ±1/2 corresponds to a site occupied or unoccupied by a Cooper pair, while superconductivity corresponds to a spontaneous in-plane magnetization, e.g. hSix i = 6 0. Disorder is represented by the random transverse field ξi , box distributed between −W and W . The optical response of classical32 and quantum33 XY -like models has been addressed previously, by introducing disorder in the coupling J. The model (8) focuses instead on the competition between pair hopping (J) and localization (W )19,34–36 , that has been recently proven successfull5,35 to describe the STM experimental results in the SC phase near the SIT. We first solve the model (8) in mean-field to determine hSix i = 12 sin θi and then rotate to the local coordinate fz = S z cos θi + system such that the new z-axis is S i i x Si sin θi . At strong disorder the SC order parameter develops an inhomogeneous spatial distribution, with SC islands embedded in an insulating background (cfr Fig. 6 below), in analogy both with the fermionic model (1)11,12,16 and with tunnelling experiments5,8 . Small fluctuations with respect to the mean-field configuration can be described by means of a Holstein-Primakov (HP) scheme, where spins are bosonized as usual as − fz = 1/2 − a+ a , Sf+ ' a and Sf S ' a+ . Here we have i

III.

X

i

i

i

i

i

i

± z x fy fx fx defined Sf i = Si ± iSi with Si = −Si sin θi + Si cos θi y y f and Si = Si being orthogonal to the local quantization axis. The Hamiltonian (8) is then mapped into a quadratic model HP S = EM F + HP0 S that can be diagonalized by means of a Bogoliubov transformation

4 P

HP0 S

α


uαi γα + vαi γα† :

 X 1 † = Aij (ai aj + h.c.) + Bij (ai aj + h.c.) 2 ij X = Eα γα† γα + const. (9) α

Here Aij = 2δij ξi / cos θi − J(1 + cos θi cos θj )(1 − δij ) and Bij = J(1 − cos θi cos θj )(1 − δij ) are the matrices that enter in the eigenvalue problem for the excitation energies Eα 37 . The equivalence between the HP excitations and the SC phase excitations at Gaussian level can be made explicit by the identification of the phase operators Φi and their conjugated momenta Li , X φαi Siy = i √ (γα† − γα ), sin θi 2 α X `αi √ (γα† + γα ), Li = Si⊥ sin θi = 2 α

Φi = −2

(10) (11)

√ where φαi = √ 2 (vαi − uαi ) / sin θi and `αi = (uαi + vαi ) sin θi / 2. The fluctuation part of the Hamiltonian (9) can then be expressed as: HP0 S =

1 X µ 1 X −1 2 Ji [∆µ Φi ] + X Li Lj 2 i,µ=x,y 2 ij ij

(12)

where Jiµ ≡ J sin θi sin θi+ˆµ are the local stiffnesses of the disordered superconductor, ∆µ is the discrete derivative in the µ direction and Xij−1 = 2(Aij + Bij )/ sin θi sin θj is the inverse matrix of the compressibilities. Consistently with the identification (10) the usual Peierls coupling to the gauge field in the pseudospin model (8) corre− − sponds to the replacement Si+ Si+µ → Si+ Si+µ e−2ieAµ , with a factor of 2 accounting for the double charge of each Cooper pair. This leads in Eq. (12) to the shift ∆µ Φi → ∆µ Φi − 2eAµ , i.e. the usual minimalcoupling scheme. The real part of the optical conductivity for the bosonic model (8) is then easily obtained as B σ B (ω) = e2 πδ(ω)DsB + σreg (ω) with DsB = DB − B σreg (ω) =

1 X Zα N α

(13)

e2 π X Zα [δ(ω + Eα ) + δ(ω − Eα )] ,(14) 2N α

P where DB = (1/N ) i 4Jiµ is the diamagnetic term of the bosonic model (8), µ = x for instance and the effective dipole Zα of each excitation mode is " #2 1 X µ Zα = 2Ji ∆µ φαi . Eα i

(15)

For a uniform stiffness (Jiµ =const) one finds that Zα

0.12 0.1 σ(ω)/σ0

ai =

0.08

W/J=3 W/J=5.6 W/J=8.5 W/J=11 W/J=18 W/J=24

4

B

2

0.6 Optical weight

1 0.06

0.8

D /J B D s/J

3

0

0.4 0.2 0

0 5 10 15 20 25 30 W/J

0.04 0.02 0

0

5

10

ω/J

15

20

FIG. 4: σreg (ω)/σ0 for the bosonic model (8) at different values of W/J. The lattice size is N = 50 × 50 and the average is taken over 100 disorder configurations. Inset: disorder dependence of the diamagnetic term DRB , the superfluid ∞ stiffness DsB and the total spectral weight 0+ dωσreg (ω) = B B (π/2)(D − Ds ) in units of J.

is proportional to the total derivative of the phase modulation, which then vanishes for periodic boundary conditions. Thus, the inhomogeneity of Jiµ induced by disorder is a crucial prerequisite to obtain a finite electric dipole, responsible for the σreg (ω) shown in Fig. 4. As one can see, the optical response moves towards decreasing energies for increasing R ∞ disorder (i.e. W/J), and its total spectral weight 0+ dωσreg (ω) = (π/2)(DB − DsB ) [see Eq. (13)-(14)] first increases, due to the disordertuned optical absorption at finite ω, and then decreases again, due to the strong suppression by disorder of the diamagnetic term DB itself (see inset Fig. 4). Notice that the decrease of DB with increasing disorder reflects the suppression of the local order parameter, encoded in the fermionic language (1) in the suppression of the BCS stiffness DsBCS . This analogy can be used to obtain a quantitative comparison between the fermionic and the bosonic approach, by fixing W/J of the model (8) in order to reproduce DsB /DB = Ds /DsBCS . In this way we can account in both models for the same transfer of spectral weight from ω = 0 to σreg (ω) (see Appendix B). The results are shown in the insets of Fig. 2 and in Fig. 3: as one can see, at large U the bosonic model reproduces in a quantitative way the characteristic energy scales for optical absorption in the fermionic model. At weaker coupling the comparison is instead only qualitative, due partly to the difficulties of clearly separating the contribution of quasiparticles and collective modes. Let us finally analyze the connection between the optical response and the inhomogeneous spatial distribution of the SC properties. The optical response (14) is proportional to the density of states of phase modes N (ω), weighted by the effective dipole function Zα of Eq. (15). Both quantities depend on disorder, as it is shown in Fig.

5 WJ=10

10

5

ω/J, Eα/J

0.4

0.04

0.2

Z/J

0.6 0.2

0 15

0 0

10

20

30

ω/J, Eα/J

15

15

10

10

5

5

0 40

(c)

Eα/J=0.53, Zα/J